#8388796

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4585 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,16)(12,18)(13,14)(15,17)(19,24)(20,23)(21,22)(25,26,31,32)(27,34,28,33)(29,30,36,35)(37,38,39,40);(1,2,5,6)(3,7,8,4)(9,11,15,20)(10,16,17,23)(12,19,22,14)(13,18,24,21)(26,32)(29,36)(33,34)(37,39);(2,6)(4,7)(10,17)(11,20)(14,19)(18,21)(25,27,31,28)(26,33,32,34)(29,37,36,39)(30,40,35,38);(2,6)(3,8)(9,12)(10,18)(11,19)(13,23)(14,20)(15,22)(16,24)(17,21)(25,28,31,27)(26,34,32,33)(29,37,36,39)(30,40,35,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,19)(11,21)(12,16)(14,17)(15,24)(18,20)(22,23)(25,29,31,36)(26,35,32,30)(27,37,28,39)(33,38,34,40);(1,4)(2,3)(5,7)(6,8)(9,14,15,19)(10,13,17,24)(11,12,20,22)(16,18,23,21)(25,30,31,35)(26,36,32,29)(27,38,28,40)(33,37,34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8295992);